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Interpolating a DFT

Starting with a sampled spectrum $ X(\omega_k)$ , $ k=0,1,\ldots,N-1$ , typically obtained from a DFT, we can interpolate by taking the DTFT of the IDFT which is not periodically extended, but instead zero-padded [264]:3.8

\begin{eqnarray*}
X(\omega) &=& \hbox{\sc DTFT}(\hbox{\sc ZeroPad}_{\infty}(\hbox{\sc IDFT}_N(X)))\\
&\isdef & \sum_{n=-N/2}^{N/2-1}\left[\frac{1}{N}\sum_{k=0}^{N-1}X(\omega_k)
e^{j\omega_k n}\right]e^{-j\omega n}\\
&=& \sum_{k=0}^{N-1}X(\omega_k)
\left[\frac{1}{N}\sum_{n=-N/2}^{N/2-1} e^{j(\omega_k-\omega) n}\right]\\
&=& \sum_{k=0}^{N-1}X(\omega_k)\,\hbox{asinc}_N(\omega-\omega_k)\\
&=& \left<X,\hbox{\sc Sample}_N\{\hbox{\sc Shift}_{\omega}(\hbox{asinc}_N)\}\right>
\end{eqnarray*}

(The aliased sinc function, $ \hbox{asinc}_N(\omega)$ , is derived in §3.1.) Thus, zero-padding in the time domain interpolates a spectrum consisting of $ N$ samples around the unit circle by means of `` $ \hbox{asinc}_N$ interpolation.'' This is ideal, time-limited interpolation in the frequency domain using the aliased sinc function as an interpolation kernel. We can almost rewrite the last line above as $ X(\omega)=(X\ast \hbox{asinc}_N)_\omega$ , but such an expression would normally be defined only for $ \omega=\omega_l=2\pi l/N$ , where $ l$ is some integer, since $ X$ is discrete while $ \hbox{asinc}_N$ is continuous.

Figure F.1 lists a matlab function for performing ideal spectral interpolation directly in the frequency domain. Such an approach is normally only used when non-uniform sampling of the frequency axis is needed. For uniform spectral upsampling, it is more typical to take an inverse FFT, zero pad, then a longer FFT, as discussed further in the next section.


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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